Compactifications of phylogenetic systems and species of electrical networks

TL;DR

This paper introduces new spaces and maps linking circular electrical networks with phylogenetic split systems, extending to compactifications and preserving structures, with applications in combinatorics and phylogenetics.

Contribution

It develops a graphical map connecting electrical networks and split systems, extending to cactus networks and providing a combinatorial framework for phylogenetic analysis.

Findings

Graphical map relates electrical networks to split systems

Spaces are CW complexes with Bell number cell counts

Extension to cactus networks preserves structures

Abstract

We describe new spaces and maps. Our graphical map is a visual and numerical correspondence between spaces of circular electrical networks and circular planar split systems. When restricted to the planar circular electrical case, this graphical map finds the split system uniquely associated with the Kalmanson resistance distance of the dual network, matching the induced split system familiar from phylogenetics. This correspondence is extended to compactifications of the respective spaces, taking cactus networks to the cactus split systems defined herein. The graphical map preserves both network components and cactus structure, allowing an elegant enumeration of induced phylogenetic split systems via combinatorial species. We introduce the global spaces of circular planar electrical networks and circular split systems. These new spaces are also CW complexes, but the 0-cells of each are counted by the Bell numbers as opposed to the Catalan numbers. As species, the two sorts of global cacti are seen to be compositions in complementary ways.